The LES model, developed by Deardorff (1980) and Moeng (1984) for the atmospheric boundary layer, dynamically solves for unresolved (subgrid) turbulent kinetic energy (TKE). Several adaptations of this model are currently used to model oceanic boundary layers, including high-latitude deep convection in response to storm forcing (Harcourt et al., 2001), and forced convection in more shallow free-slip ocean boundary layers (e.g. Skyllingstad and Denbo, 1995; Wang et al, 1996).
Following Moeng (1984),
the advection terms of the momentum equations
are computed prognostically from vorticity components
Terms involving both subgrid and resolved TKE are absorbed into a
modified nonhydrostatic pressure 
. b is the buoyancy force, and the mean
acceleration is removed from the right-hand side of the vertical momentum
equation to maintain a prescribed mean vertical velocity.
The LES uses a finite series expansion of the equation of state about a
potential reference state 
with density 
chosen to
minimize truncation errors. After removing the horizontally averaged vertical
acceleration 
, buoyancy fluctuations 
account for
mean scalar dependence in the expansion coefficients, thermobaricity, cabbeling
and a ‘halobaric’ effect through 
, 
, 
and 
, as
modifications to the standard approximation of constant thermal and haline
expansion coefficients 
and 
. The salinity-related components of cabbeling and the
dependence of the haline expansion coefficient on mean salinity are small and
generally omitted.
The evolution of potential temperature is computed prognostically from
with corresponding equations for salinity and Lagrangian tracers. The velocity
components are subject to the condition of incompressibility, which is applied
to the momentum equations to solve diagnostically for 
. Unresolved
stresses are modeled in terms of the symmetric resolved strain rate and local
nonlinear eddy viscosity 
; scalar fluxes in terms of resolved gradients and eddy
diffusivity 
:
Parameterizations eddy viscosity diffusivity, 
and 
, depend upon subgrid TKE 
, its associated
dissipation length scale 
, turbulent eddy Prandtl number 
and constant
. The
length scale 
of unresolved motion is equal to the resolution scale 
, unless reduced
below that by stability or by proximity to a wall boundary. Parameterizations
for the dissipation length scale 
for stably stratified turbulence are discussed
below in the description of simulations. Resolution scale 
is governed
either explicitly by a spatial filter, implicitly by the numerical scheme, or
by a combination of the two.
The subgrid TKE equation
includes unresolved buoyant production 
as well as
unresolved shear production and transport, and parameterizes dissipation as
. When a
portion of the turbulent inertial subrange is resolved, constants 
and 
of the subgrid parameterization are determined by integrals
over the 
Kolmogorov energy spectrum for unresolved TKE smaller than
